Video Transcript
Consider the function 𝑓 of 𝑥 is
equal to log base two of three 𝑥 minus one. If 𝑓 of 𝑎 is equal to three, find
the value of 𝑎.
We are told that 𝑓 of 𝑎 is equal
to three, so we can begin by substituting these values into the function 𝑓 of
𝑥. This gives us log base two of three
𝑎 minus one is equal to three. We recall that logarithmic
functions are the inverses of exponential functions. This means that if log base 𝑎 of
𝑦 is equal to 𝑥, then 𝑎 to the power of 𝑥 is equal to 𝑦. In this question, the base 𝑎 is
equal to two, the variable 𝑦 is equal to three 𝑎 minus one, and the variable 𝑥 is
equal to three. Two cubed is therefore equal to
three 𝑎 minus one.
We know that two cubed is equal to
eight. We can then add one to both sides
of this equation so that three 𝑎 is equal to nine. Dividing both sides of this
equation by three gives us 𝑎 is equal to three. If the function 𝑓 of 𝑥 is equal
to log base two of three 𝑥 minus one and 𝑓 of 𝑎 is equal to three, then the value
of 𝑎 is three. We could check this answer on the
calculator by substituting our value back in to the original function.
